Maximum product fuzzy relational inequalities with fuzzy constraints: an exact algorithm

Document Type : Research Paper

Authors

1 Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran.

2 Faculty of Engineering Science, College of Engineering, University of Tehran.

Abstract

Fuzzy relational inequalities with fuzzy constraints (FRI-FC) represent a generalized form of fuzzy relational inequalities (FRI), in which fuzzy inequality replaces ordinary inequality in the constraints. With fuzzy constraints, we can obtain optimal points (called super-optima) that provide better solutions than those obtained by addressing similar problems with ordinary inequality constraints. In this paper, a linear objective function optimization problem with respect to a max-product FRI-FC problem is considered. Several optimization problems are equivalent to the primal problem, as it has been proven. The main problem is converted into a more simplified problem by means of some simplification operations based on the algebraic structure of the primary problem and its equivalent forms. As a result of a few mathematical manipulations, the main problem has been transformed into a linear optimization model. A super-optimum (which is better than the ordinary feasible optimal solution) is not only found by applying the proposed linearization method, but the best super-optimum is also identified for the main problem. A comparison is made between the current approach and our previous work, in addition to some well-known heuristic algorithms, by applying them to random test problems of different sizes and seeing how they compare to each other. Results demonstrate that the proposed method produces optimal solutions with admissible infeasibilities, while the linearization algorithm produces better solutions than other heuristic algorithms. In addition, the results demonstrate that heuristic algorithms could not escape from poor solutions in most cases.

Keywords

Main Subjects

References

[1] C. W. Chang, B. S. Shieh, Linear optimization problem constrained by fuzzy max–min relation equations, Information
Sciences, 234 (2013), 71-79. https://doi.org/10.1016/j.ins.2011.04.042
[2] L. Chen, P. P. Wang, Fuzzy relation equations (I): The general and specialized solving algorithms, Soft Computing,
6(6) (2002), 428-435. https://doi.org/10.1007/s00500-001-0157-3
[3] L. Chen, P. P. Wang, Fuzzy relation equations (II): The branch-point-solutions and the categorized minimal solutions,
Soft Computing, 11(1) (2007), 33-40. https://doi.org/10.1007/s00500-006-0050-1
[4] S. Dempe, A. Ruziyeva, On the calculation of a membership function for the solution of a fuzzy linear optimization
problem, Fuzzy Sets and Systems, 188(1) (2012), 58-67. https://doi.org/10.1016/j.fss.2011.07.014
[5] A. Di Nola, E. Sanchez, W. Pedrycz, S. Sessa, Fuzzy relation equations and their applications to knowledge engineering,
Springer Dordrecht, 1989. https://doi.org/10.1007/978-94-017-1650-5
[6] D. Dubey, S. Chandra, A. Mehra, Fuzzy linear programming under interval uncertainty based on IFS representation,
Fuzzy Sets and Systems, 188(1) (2012), 68-87. https://doi.org/10.1016/j.fss.2011.09.008
[7] Y. R. Fan, G. H. Huang, A. L. Yang, Generalized fuzzy linear programming for decision making under uncertainty:
Feasibility of fuzzy solutions and solving approach, Information Sciences, 241 (2013), 12-27. https://doi.org/
10.1016/j.ins.2013.04.004
[8] S. C. Fang, G. Li, Solving fuzzy relation equations with a linear objective function, Fuzzy Sets and Systems, 103(1)
(1999), 107-113. https://doi.org/10.1016/s0165-0114(97)00184-x
[9] S. Freson, B. De Baets, H. De Meyer, Linear optimization with bipolar max–min constraints, Information Sciences,
234 (2013), 3-15. https://doi.org/10.1016/j.ins.2011.06.009
[10] Z. W. Geem, J. H. Kim, G. V. Loganathan, A new heuristic optimization algorithm: Harmony search, Simulation,
76(2) (2001), 60-68. https://doi.org/10.1177/003754970107600201
[11] A. Ghodousian, An algorithm for solving linear optimization problems subjected to the intersection of two fuzzy
relational inequalities defined by Frank family of t-norms, International Journal in Foundations of Computer Science
and Technology, 8(3) (2018), 1-20. https://doi.org/10.5121/ijfcst.2018.8301
[12] A. Ghodousian, Optimization of linear problems subjected to the intersection of two fuzzy relational inequalities
defined by Dubois-Prade family of t-norms, Information Sciences, 503 (2019), 291-306. https://doi.org/10.
1016/j.ins.2019.06.058
[13] A. Ghodousian, A. Babalhavaeji, An efficient genetic algorithm for solving nonlinear optimization problems defined
with fuzzy relational equations and max- Lukasiewicz composition, Applied Soft Computing, 69 (2018), 475-492.
https://doi.org/10.1016/j.asoc.2018.04.029
[14] A. Ghodousian, M. S. Chopannavaz, Solving linear optimization problems subject to bipolar fuzzy relational equalities
defined with max-strict compositions, Information Sciences, 650 (2023), 119696. https://doi.org/10.1016/
j.ins.2023.119696
[15] A. Ghodousian, M. S. Chopannavaz, On the nonlinear programming problems subject to a system of generalized
bipolar fuzzy relational equalities defined with continuous t-norms, Iranian Journal of Fuzzy Systems, 22(3) (2025),
21-38. https://doi.org/10.22111/ijfs.2025.50433.8905
[16] A. Ghodousian, M. S. Chopannavaz, W. Pedrycz, On the resolution and linear optimization problems subject to a
system of bipolar fuzzy relational equalities defined with continuous Archimedean t-norms, Iranian Journal of Fuzzy
Systems, 22(4) (2025), 137-160. https://doi.org/10.22111/ijfs.2025.50906.8994
[17] A. Ghodousian, E. Khorram, Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints
with max–min composition, Information Sciences, 178(2) (2008), 501-519. https://doi.org/10.1016/j.ins.
2007.07.022
[18] A. Ghodousian, E. Khorram, Linear optimization with an arbitrary fuzzy relational inequality, Fuzzy Sets and
Systems, 206 (2012), 89-102. https://doi.org/10.1016/j.fss.2012.04.009
[19] A. Ghodousian, M. Naeeimi, A. Babalhavaeji, Nonlinear optimization problem subjected to fuzzy relational equations
defined by Dubois-Prade family of t-norms, Computers and Industrial Engineering, 119 (2018), 167-180. https:
//doi.org/10.1016/j.cie.2018.03.038
[20] A. Ghodousian, M. R. Parvari, A modified PSO algorithm for linear optimization problem subject to the generalized
fuzzy relational inequalities with fuzzy constraints (FRI-FC), Information Sciences, 418 (2017), 317-345. https:
//doi.org/10.1016/j.ins.2017.07.032
[21] A. Ghodousian, R. Rabie, T. Azarnejad, Solving a non-linear optimization problem constrained by a non-convex
region defined by fuzzy relational equations and Schweizer-Sklar family of T-norms, American Journal of Computation,
Communication and Control, 5(2) (2018), 68-87.
[22] F. F. Guo, L. P. Pang, D. Meng, Z. Q. Xia, An algorithm for solving optimization problems with fuzzy relational
inequality constraints, Information Sciences, 252 (2013), 20-31. https://doi.org/10.1016/j.ins.2011.09.030
[23] F. F. Guo, Z. Q. Xia, An algorithm for solving optimization problems with one linear objective function and finitely
many constraints of fuzzy relation inequalities, Fuzzy Optimization and Decision Making, 5(1) (2006), 33-47.
https://doi.org/10.1007/s10700-005-4914-0
[24] S. M. Guu, Y. K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy
Optimization and Decision Making, 1(4) (2002), 347-360. https://doi.org/10.1023/A:1020955112523
[25] S. M. Guu, Y. K. Wu, Minimizing a linear objective function under a max-t-norm fuzzy relational equation constraint,
Fuzzy Sets and Systems, 16(2) (2010), 285-297. https://doi.org/10.1016/j.fss.2009.03.007
[26] K. Hirota, W. Pedrycz, Solving fuzzy relational equations through logical filtering, Fuzzy Sets and Systems, 81(3)
(1996), 355-363. https://doi.org/10.1016/0165-0114(95)00221-9
[27] J. Kennedy, R. Eberhart, Particle swarm optimization, IEEE Proceedings of ICNN’95-International Conference on
Neural Networks, 4 (1995), 1942-1948. https://doi.org/10.1007/978-0-387-30164-8_630
[28] P. Li, S. C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-
T composition, Fuzzy Optimization and Decision Making, 7(2) (2008), 169-214. https://doi.org/10.1007/
s10700-008-9029-y
[29] P. Li, Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using the  Lukasiewicz triangular
norm, Soft Computing, 18(7) (2014), 1399-1404. https://doi.org/10.1007/s00500-013-1152-1
[30] J. X. Li, S. Yang, Fuzzy relation inequalities about the data transmission mechanism in BitTorrent-like Peer-to-Peer
file sharing systems, IEEE 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery, (2012),
452-456. https://doi.org/10.1109/fskd.2012.6233956
[31] J. L. Lin, On the relation between fuzzy max-archimedean t-norm relational equations and the covering problem,
Fuzzy Sets and Systems, 160(16) (2009), 2328-2344. https://doi.org/10.1016/j.fss.2009.01.012
[32] J. L. Lin, Y. K. Wu, S. M. Guu, On fuzzy relational equations and the covering problem, Information Sciences,
181(14) (2011), 2951-2963. https://doi.org/10.1016/j.ins.2011.03.004
[33] C. C. Liu, Y. Y. Lur, Y. K. Wu, Linear optimization of bipolar fuzzy relational equations with max- Lukasiewicz
composition, Information Sciences, 360 (2016), 149-162. https://doi.org/10.1016/j.ins.2016.04.041
[34] J. Loetamonphong, S. C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets
and Systems, 118(3) (2001), 509-517. https://doi.org/10.1016/s0165-0114(98)00417-5
[35] A. V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy
Sets and Systems, 153(2) (2005), 261-273. https://doi.org/10.1016/j.fss.2005.02.010
[36] W. Pedrycz, Fuzzy relational equations with generalized connectives and their applications, Fuzzy Sets and Systems,
10(1-3) (1983), 185-201. https://doi.org/10.1016/s0165-0114(83)80114-6
[37] K. Price, R. M. Storn, J. A. Lampinen, Differential evolution: A practical approach to global optimization (natural
computing series), Springer-Verlag, 2005. https://doi.org/540-31306-0_2
[38] X. B. Qu, X. P. Wang, Minimization of linear objective functions under the constraints expressed by a system
of fuzzy relation equations, Information Sciences, 178(17) (2008), 3482-3490. https://doi.org/10.1016/j.ins.
2008.04.004
[39] X. B. Qu, X. P. Wang, M. H. Lei, Conditions under which the solution sets of fuzzy relational equations over
complete Brouwerian lattices form lattices, Fuzzy Sets and Systems, 234 (2014), 34-45. https://doi.org/10.
1016/j.fss.2013.03.017
[40] E. Sanchez, Solutions in composite fuzzy relations equations-application to medical diagnosis in Brouwerian logic,
Fuzzy Automated and Decision Processes, (1977), 221-234. https://doi.org/10.1016/b978-1-4832-1450-4.
50017-1
[41] E. Sanchez, Resolution of Eigen fuzzy sets equations, Fuzzy Sets and Systems, 1(1) (1978), 69-74. https://doi.
org/10.1016/0165-0114(78)90033-7
[42] B. S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Information Sciences, 178(8) (2008), 1961-
1967. https://doi.org/10.1016/j.ins.2007.12.006
[43] B. S. Shieh, Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint, Information
Sciences, 181(4) (2011), 832-841. https://doi.org/10.1016/j.ins.2010.10.024
[44] K. Socha, M. Dorigo, Ant colony optimization for continuous domains, European Journal of Operational Research,
185(3) (2008), 1155-1173. https://doi.org/10.1016/j.ejor.2006.06.046
[45] F. Sun, Conditions for the existence of the least solution and minimal solutions to fuzzy relation equations over
complete Brouwerian lattices, Information Sciences, 205 (2012), 86-92. https://doi.org/10.1016/j.ins.2012.
04.002
[46] F. Sun, X. P. Wang, X. B. Qu, Minimal join decompositions and their applications to fuzzy relation equations
over complete Brouwerian lattices, Information Sciences, 224 (2013), 143-151. https://doi.org/10.1016/j.ins.
2012.10.038
[47] Y. K. Wu, Optimization of fuzzy relational equations with max-av composition, Information Sciences, 177(19)
(2007), 4216-4229. https://doi.org/10.1016/j.ins.2007.02.037
[48] Y. K. Wu, S. M. Guu, Minimizing a linear function under a fuzzy max–min relational equation constraint, Fuzzy
Sets and Systems, 150(1) (2005), 147-162. https://doi.org/10.1016/j.fss.2004.09.010
[49] Y. K. Wu, S. M. Guu, An efficient procedure for solving a fuzzy relational equation with max–archimedean t-norm
composition, IEEE Transactions on Fuzzy Systems, 16(1) (2008), 73-84. https://doi.org/10.1109/tfuzz.2007.
902018
[50] Y. K. Wu, S. M. Guu, J. Y. C. Liu, Reducing the search space of a linear fractional programming problem under
fuzzy relational equations with max-Archimedean t-norm composition, Fuzzy Sets and Systems, 159(24) (2008),
3347-3359. https://doi.org/10.1016/j.fss.2008.04.007
[51] Q. Q. Xiong, X. P. Wang, Fuzzy relational equations on complete Brouwerian lattices, Information Sciences, 193
(2012), 141-152. https://doi.org/10.1016/j.ins.2011.12.030
[52] S. J. Yang, An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with
addition–min composition, Fuzzy Sets and Systems, 255 (2014), 41-51. https://doi.org/10.1016/j.fss.2014.
04.007
[53] X. P. Yang, X. G. Zhou, B. Y. Cao, Latticized linear programming subject to max-product fuzzy relation inequalities
with application in wireless communication, Information Sciences, 358 (2016), 44-55. https://doi.org/10.1016/
j.ins.2016.04.014
Iranian Journal of Fuzzy Systems
Volume 23, Issue 1
January and February 2026
Pages 1-21

Files

Share

How to cite

Statistics